Polynomial Curves in Euclidean Space

• Polynomial Parametric Curves
• Lagrange Interpolation
  • One-dimensional Example
  • Neville’s Algorithm
  • Two-dimensional Example
  • Runge’s Phenomenon
• Hermite Splines
  • Properties
  • Derivation (Uniform)
    • Relation to Bézier Splines
  • Derivation (Non-Uniform)
• Natural Splines
  • Properties
  • Derivation (Uniform)
    • End Conditions
      • Natural
      • Clamped
      • Closed
    • Solving the System of Equations
  • Derivation (Non-Uniform)
    • End Conditions
      • Natural
      • Other End Conditions
• Bézier Splines
  • Properties
  • De Casteljau's Algorithm
    • Preparations
    • Degree 1, a.k.a. linear
    • Degree 2, a.k.a. quadratic
    • Degree 3, a.k.a. cubic
    • Degree 4, a.k.a. quartic
    • Arbitrary Degree
• Catmull--Rom Splines
  • Properties
    • Tangent Vectors
    • Cusps and Self-Intersections
    • Chordal Parameterization
    • Centripetal Parameterization
    • Parameterized Parameterization
  • Derivation (Uniform)
    • Blending Functions
    • Cardinal Functions
    • Example Plot
    • Basis Polynomials
    • Basis Matrix
    • Tangent Vectors
  • Derivation (Non-Uniform)
    • Tangent Vectors
    • Basis Matrix
  • Barry--Goldman Algorithm
    • Triangular Schemes
    • Neville’s Algorithm
    • De Boor’s Algorithm
    • Combining Both Algorithms
    • Step by Step
      • First Stage
      • Second Stage
      • Third Stage
    • Tangent Vectors
    • Animation
• Kochanek--Bartels Splines
  • Properties
    • Tension
    • Continuity
    • Bias
    • Combinations
  • Derivation (Uniform)
    • Parameters
      • Tension
      • Continuity
      • Bias
      • All Three Combined
    • Calculation
  • Derivation (Non-Uniform)
    • Parameters
    • Calculation
• End Conditions
  • Natural End Conditions
    • Begin
    • End
    • Example
• Piecewise Monotone Interpolation
  • Examples
    • Providing Slopes
  • Generating and Modifying the Slopes at Segment Boundaries
  • PCHIP/PCHIM
  • More Examples
  • Monotone Interpolation
  • End Conditions
  • Even More Examples